Best season to move to Indonesia

What is the best season/time of the year to move to Indonesia, and why?

Not really

How do rent prices fluctuate based on the different seasons?

Except (possibly) in the holiday areas, they don't much.

Is there a change in the job market depending on the seasons, such as job availability?

This depends on many factors but, for those intending to work as teachers, contracts usually start in July as that's the start of the new year.
Other employers could want you to start at any time.

Are there any seasonal festivals or cultural events that would allow you to learn about the country?

A lot, far too many to mention but the major religious holidays are fixed to the international calendar.
Some are massive, the Muslim Ramadan fasting month being easily the biggest but Christmas is a big one as well.

What are the times of year to avoid for expatriation to Indonesia?

Ramadan is probably best avoided for new expats from non-Muslim countries as you're very likely to commit a cultural faux pas or several.
Also, I would strongly advise avoiding turning up in Bali on their day of silence as unwitting new expats could very well cause a stir.

During which season did you move? How did it go?

Ramadan, and it hard work that first year.